Subject: [mg109616] Number of ways of permutations to form a certain pattern of cycles?

From: newlearner <poincare100 at gmail.com>

Date: Sat, 8 May 2010 07:07:16 -0400 (EDT)

Dear all,
How to compute in Mathematica the number of ways of permutations of n-
objects to form a certain pattern of cycles?
I mean, for example, let n = 3. then there are 6 ways of permutations
of three objects
123, 231, 312, 132, 213, 321
Among them, we have the three 1-cycle pattern, 123
two 3-cycle pattern, 231, 312
one 1-cycle and one 2-cycle pattern, 132, 213, 321
How to calculate this kind of mathematical problem with Mathematica?
Could you also show me how to calculate the number of ways without
using of Mathematica?
Thanks so much!